Theorem elmpps 35578

Description: Definition of a provable pre-statement, essentially just a reorganization of the arguments of df-mcls . (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

Ref	Expression
mppsval.p	⊢ 𝑃 = (mPreSt‘𝑇)
mppsval.j	⊢ 𝐽 = (mPPSt‘𝑇)
mppsval.c	⊢ 𝐶 = (mCls‘𝑇)

Assertion

Ref	Expression
elmpps	⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝐽 ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐴 ∈ (𝐷𝐶𝐻)))

Proof of Theorem elmpps

Dummy variables 𝑎 𝑑 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ot 4635	. . 3 ⊢ ⟨𝐷, 𝐻, 𝐴⟩ = ⟨⟨𝐷, 𝐻⟩, 𝐴⟩
2		mppsval.p	. . . 4 ⊢ 𝑃 = (mPreSt‘𝑇)
3		mppsval.j	. . . 4 ⊢ 𝐽 = (mPPSt‘𝑇)
4		mppsval.c	. . . 4 ⊢ 𝐶 = (mCls‘𝑇)
5	2, 3, 4	mppsval 35577	. . 3 ⊢ 𝐽 = {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))}
6	1, 5	eleq12i 2834	. 2 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝐽 ↔ ⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))})
7		oprabss 7541	. . . 4 ⊢ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))} ⊆ ((V × V) × V)
8	7	sseli 3979	. . 3 ⊢ (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))} → ⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ ((V × V) × V))
9	2	mpstssv 35544	. . . . . 6 ⊢ 𝑃 ⊆ ((V × V) × V)
10	9	sseli 3979	. . . . 5 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → ⟨𝐷, 𝐻, 𝐴⟩ ∈ ((V × V) × V))
11	1, 10	eqeltrrid 2846	. . . 4 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → ⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ ((V × V) × V))
12	11	adantr 480	. . 3 ⊢ ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐴 ∈ (𝐷𝐶𝐻)) → ⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ ((V × V) × V))
13		opelxp 5721	. . . 4 ⊢ (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ ((V × V) × V) ↔ (⟨𝐷, 𝐻⟩ ∈ (V × V) ∧ 𝐴 ∈ V))
14		opelxp 5721	. . . . 5 ⊢ (⟨𝐷, 𝐻⟩ ∈ (V × V) ↔ (𝐷 ∈ V ∧ 𝐻 ∈ V))
15		simp1 1137	. . . . . . . . . 10 ⊢ ((𝑑 = 𝐷 ∧ ℎ = 𝐻 ∧ 𝑎 = 𝐴) → 𝑑 = 𝐷)
16		simp2 1138	. . . . . . . . . 10 ⊢ ((𝑑 = 𝐷 ∧ ℎ = 𝐻 ∧ 𝑎 = 𝐴) → ℎ = 𝐻)
17		simp3 1139	. . . . . . . . . 10 ⊢ ((𝑑 = 𝐷 ∧ ℎ = 𝐻 ∧ 𝑎 = 𝐴) → 𝑎 = 𝐴)
18	15, 16, 17	oteq123d 4888	. . . . . . . . 9 ⊢ ((𝑑 = 𝐷 ∧ ℎ = 𝐻 ∧ 𝑎 = 𝐴) → ⟨𝑑, ℎ, 𝑎⟩ = ⟨𝐷, 𝐻, 𝐴⟩)
19	18	eleq1d 2826	. . . . . . . 8 ⊢ ((𝑑 = 𝐷 ∧ ℎ = 𝐻 ∧ 𝑎 = 𝐴) → (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ↔ ⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃))
20	15, 16	oveq12d 7449	. . . . . . . . 9 ⊢ ((𝑑 = 𝐷 ∧ ℎ = 𝐻 ∧ 𝑎 = 𝐴) → (𝑑𝐶ℎ) = (𝐷𝐶𝐻))
21	17, 20	eleq12d 2835	. . . . . . . 8 ⊢ ((𝑑 = 𝐷 ∧ ℎ = 𝐻 ∧ 𝑎 = 𝐴) → (𝑎 ∈ (𝑑𝐶ℎ) ↔ 𝐴 ∈ (𝐷𝐶𝐻)))
22	19, 21	anbi12d 632	. . . . . . 7 ⊢ ((𝑑 = 𝐷 ∧ ℎ = 𝐻 ∧ 𝑎 = 𝐴) → ((⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ)) ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐴 ∈ (𝐷𝐶𝐻))))
23	22	eloprabga 7542	. . . . . 6 ⊢ ((𝐷 ∈ V ∧ 𝐻 ∈ V ∧ 𝐴 ∈ V) → (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))} ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐴 ∈ (𝐷𝐶𝐻))))
24	23	3expa 1119	. . . . 5 ⊢ (((𝐷 ∈ V ∧ 𝐻 ∈ V) ∧ 𝐴 ∈ V) → (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))} ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐴 ∈ (𝐷𝐶𝐻))))
25	14, 24	sylanb 581	. . . 4 ⊢ ((⟨𝐷, 𝐻⟩ ∈ (V × V) ∧ 𝐴 ∈ V) → (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))} ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐴 ∈ (𝐷𝐶𝐻))))
26	13, 25	sylbi 217	. . 3 ⊢ (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ ((V × V) × V) → (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))} ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐴 ∈ (𝐷𝐶𝐻))))
27	8, 12, 26	pm5.21nii 378	. 2 ⊢ (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))} ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐴 ∈ (𝐷𝐶𝐻)))
28	6, 27	bitri 275	1 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝐽 ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐴 ∈ (𝐷𝐶𝐻)))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  Vcvv 3480  ⟨cop 4632  ⟨cotp 4634   × cxp 5683  ‘cfv 6561  (class class class)co 7431  {coprab 7432  mPreStcmpst 35478  mClscmcls 35482  mPPStcmpps 35483

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-ot 4635  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpst 35498  df-mpps 35503

This theorem is referenced by:  mthmpps  35587  mclspps  35589